|
%I #21 Oct 05 2018 08:21:25
%S 3,6,6,12,36,126,135,198,630,642,762,1950,1965,2160,4680,4698,4986,
%T 9576,9597,9996,17556,17580,18108,29700,29727,30402,47250,47280,48120,
%U 71610,71643,72666,104346,104382,105606,147186,147225,148668,202020,202062,203742
%N a(n) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + ... + (up to the n-th term).
%C For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=3.
%H Colin Barker, <a href="/A319867/b319867.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).
%F From _Colin Barker_, Sep 30 2018: (Start)
%F G.f.: 3*x*(1 + x - 2*x^3 + 4*x^4 + 30*x^5 + x^6 - 5*x^7 + 24*x^8) / ((1 - x)^5*(1 + x + x^2)^4).
%F a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n>13.
%F (End)
%e a(1) = 3;
%e a(2) = 3*2 = 6;
%e a(3) = 3*2*1 = 6;
%e a(4) = 3*2*1 + 6 = 12;
%e a(5) = 3*2*1 + 6*5 = 36;
%e a(6) = 3*2*1 + 6*5*4 = 126;
%e a(7) = 3*2*1 + 6*5*4 + 9 = 135;
%e a(8) = 3*2*1 + 6*5*4 + 9*8 = 198;
%e a(9) = 3*2*1 + 6*5*4 + 9*8*7 = 630;
%e a(10) = 3*2*1 + 6*5*4 + 9*8*7 + 12 = 642;
%e a(11) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11 = 762;
%e a(12) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 = 1950;
%e a(13) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15 = 1965;
%e a(14) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14 = 2160;
%e a(15) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 = 4680;
%e a(16) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18 = 4698;
%e a(17) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17 = 4986;
%e a(18) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17*16 = 9576;
%e a(19) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17*16 + 21 = 9597;
%e etc.
%p a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,3),n=1..45); # _Muniru A Asiru_, Sep 30 2018
%t k:=3; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k]) * Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k]) * Product[j-i+1, {i,1,k}], {j,1,n}]; Array[a, 50] (* _Stefano Spezia_, Sep 30 2018 *)
%o (PARI) Vec(3*x*(1 + x - 2*x^3 + 4*x^4 + 30*x^5 + x^6 - 5*x^7 + 24*x^8) / ((1 - x)^5*(1 + x + x^2)^4) + O(x^50)) \\ _Colin Barker_, Sep 30 2018
%Y For similar sequences, see: A000217 (k=1), A319866 (k=2), this sequence (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).
%Y Cf. A268685 (trisection).
%K nonn,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 29 2018
|
